Done if by Hand or Hew if by Sword?

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Andúril, the Flame of the West, forged from the shards of Narsil (and that sword fashioned by the dwarven hands of Telchar), inspired friends and frightened foes. With this sword Aragorn wins the Battle of Helm's Deep, breaks the Siege of Minas Tirith and assaults the Black Gate of Mordor (with help).

So the sword and the hand that wielded it saw much work (and blood) in the closing days of the Third Age.

But what did Aragorn mean, when laying his sword beside the door of Meduseld at King Théoden's command, that:

Originally Posted by Aragorn

"But I command you not to touch it, nor to permit any other to lay hand on it....Death shall come to any man that draws Elendil's sword, save Elendil's heir."

Did Aragorn mean that the sword would not permit anyone's touch save one from his lineage, or did Aragorn mean to execute anyone that touched his sword?

Whichever, then the next question would be, why? Sure, I want to keep the kids out of my stuff, but if Aragorn meant the latter, why so severe a punishment?